When a car turns, it moves so that the rear axle lies along the radius of the circle described by the rear wheels. The front wheels also follow circles with the same centre. As a result, the paths of the four wheels lie along four distinct concentric circles, as shown:
Some definitions
The turning circle of a car is defined to be the diameter of the largest of these four circles.
The wheel base is the distance between the centres of the front and rear wheels.
The width in this context is the distance between the centres of the two rear wheels or the two front wheels. (The front and rear widths are assumed to be the same here, although this isn't necessarily always the case.)
Consider a particular car with a wheelbase of 3m and width of 2m with a turning circle of 10m. Let point M be the midpoint of the rear axle of the car. What is the diameter of the smallest circle that M can move along?
When parallel parking, point M follows an S shape where it first moves (in reverse) along an arc with full left steering lock (i.e. turning as tightly as possible) then midway through the manouver the steering is reversed so that it moves along a similar arc with full right steering lock. The end result is that the car is pointing in the same direction it started, but it has moved back and to the left. (See the diagram.)
(a) If our car (as described above) is positioned 3.3 metres away from the curb at the beginning of the manouver, how far will it have moved backward if at the end of the manouver it is 0.3m from the curb?
(b) Imagine you are the driver of the car. How would you be able to judge when to change from full left lock to full right lock?
Now generalise. Let t represent the car's smallest turning circle. Let b represent the car's wheel base. Let w represent the width. Repeat the problems above in terms of t, b and w.
Skills Needed
The mathematics this problem requires has been covered by the end of year 10. No new skills are required other than the ability to apply this mathematics in a complex situation.
Solutions
Create a page for your solution here by putting the problem title and your name in square brackets like this:
Then save this page and click on the new link you have created to go to the new page. Edit that page and put your solution there. Sign it at the bottom with four tildes (i.e. ~). This will automatically turn into a date-stamped signature like this: - glenprideaux Jan 15, 2009
For help editing these pages, see Editing.
This page has been edited 5 times. The last modification was made by - glenprideaux on Jan 15, 2009 3:08 pm
3A/3B Mathematics Specialist NPR-Parallel Parking
Parallel Parking
When a car turns, it moves so that the rear axle lies along the radius of the circle described by the rear wheels. The front wheels also follow circles with the same centre. As a result, the paths of the four wheels lie along four distinct concentric circles, as shown:
Some definitions
(a) If our car (as described above) is positioned 3.3 metres away from the curb at the beginning of the manouver, how far will it have moved backward if at the end of the manouver it is 0.3m from the curb?
(b) Imagine you are the driver of the car. How would you be able to judge when to change from full left lock to full right lock?
Skills Needed
The mathematics this problem requires has been covered by the end of year 10. No new skills are required other than the ability to apply this mathematics in a complex situation.Solutions
Create a page for your solution here by putting the problem title and your name in square brackets like this:Then save this page and click on the new link you have created to go to the new page. Edit that page and put your solution there. Sign it at the bottom with four tildes (i.e. ~). This will automatically turn into a date-stamped signature like this: -
For help editing these pages, see Editing.
This page has been edited 5 times. The last modification was made by
-